The first solution was proposed in Ref. [1]}; see also Ref. [2]}.
In Ref. [1]} it was proposed that the vector-like quarks can decay
into the SM quarks and a bosonic field \(X\) , which is automatically stable and a candidate for the cold dark matter in the Universe if it does not acquire a vacuum expectation value.
This condition defines the three different scenarios that we study in detail in this article.
